We also consider the vibrational partition function of C*. It has 3N6 (or
3 N5) terms in it, but one of those terms represents the vibration that tracks
the movement of C* into products. The high-temperature-limit partition
function for this term is (from equation 18.20)q*
hk
T
*
Let us remove this one term from the overall partition function for C* and rep-
resent the leftover partition function (minus this one vibrational term and the
electronic term) as q. We have, for Kc*:Kc*
(q (^) A/
q
V
C
)
*
(/
qV
(^) B/V)
e*/kT
h
k
T
*
(20.77)
Combining this expression with the expression for the rate constant k*, equa-
tion 20.73, we can get an expression for the rate constant kfor the bimolecu-
lar reaction:k
c°
*
(q (^) A/
q
V
C
)
*
(/
qV
(^) B/V)
e*/kT
h
k
T
*
The * terms cancel and, ifis presumed to be 1, we getk
ck
°T
h(q (^) A/
q
V
C
)
*
(/
qV
(^) B/V)
e/kT (20.78)
This last equation is one form of the Eyring equation,†and is used to estimate
kfor a bimolecular elementary process. (Careful about the two uses of the vari-
able kin the above equation!) For a given set of reactants A and B and a stated
temperature, all of the quantities in equation 20.78 can be calculated for a given
structure of a transition stateC. Therefore, if you had a known or proposed
transition state, you would be able to calculate and qfor the transition
state C, and for given reactants A and B you should certainly be able to de-
termine the partition functions (since they are typically known, stable mole-
cules). All other parameters are fundamental constants, so the rate constant k
can be calculated.
Note the parallel between equation 20.78 and the Arrhenius equation
kAeEA/RT
According to transition-state theory, the pre-exponential factor Ais given by
the expression
A
c
k
°
T
h(q (^) A/
q
V
C
)
*
(/
qV
(^) B/V)
(20.79)
Thus, we have an opportunity to calculate Afrom theoretical perspectives and
compare it to experimentally determined values (as determined, for example,
by graphing ln kversus 1/T). Table 20.2 lists some simple reactions and their
experimental and calculated pre-exponential factors. Agreements are typically
about the correct order of magnitude.
Because of the relationship between an equilibrium constant and the Gibbs
free energy, we can use K to define a G value for the formation of the tran-
722 CHAPTER 20 Kinetics
†It is named after Henry Eyring, a twentieth-century chemist who did some fundamen-
tal work in kinetics, including enunciation of the steady-state theory.